Simplify: $\frac{1}{\sqrt{6}-\sqrt{5}}-\frac{3}{\sqrt{5}-\sqrt{2}}-\frac{4}{\sqrt{6}+\sqrt{2}}$

The values of the remainder $r$,when a positive integer $a$ is divided by $3$ are $0$ and $1$ only. Justify your answer.

If $a$ and $b$ are two odd positive integers such that $a > b,$ then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and other is even.

Every odd integer $a$ is in the form of $\ldots \ldots \ldots \ldots$ where,$k \in Z$.

Write whether the square of any positive integer can be of the form $3m + 2$,where $m$ is a natural number. Justify your answer.

Prove that the following number is irrational: $\sqrt{5}-\sqrt{3}$.